Simulations and observation of nonlinear waves on the continental shelf: KdV solutions

Numerical solutions of the Korteweg-de Vries (KdV) and extended Korteweg-de Vries (eKdV) equations are used to model the transformation of a sinusoidal internal tide as it propagates across the continental shelf. The ocean is idealized as being a two-layer fluid, justified by the fact that most of the oceanic internal wave signal is contained in the gravest mode. The model accounts for nonlinear and 10 dispersive effects but neglects friction, rotation, and mean shear. The KdV model is run for a variety of idealized stratifications and unique realistic topographies to study the role of the nonlinear and dispersive effects. In all model solutions the internal tide steepens forming a sharp front from which a packet of nonlinear solitary-like waves evolves. Comparisons between KdV and eKdV solutions is explored. The model results for realistic topography and stratification are compared with observations made at moorings 15 off Massachusetts in the Middle Atlantic Bight. Some features of the observations compare well with the model. The leading face of the internal tide steepens to form a shock like front, while nonlinear high frequency waves evolve shortly after the appearance of the jump. Although not rank ordered, the wave of maximum amplitude is always close to the jump. Some features of the observations are not found in the model. Nonlinear waves can be very widely spaced and persist over a tidal period. 20 1 Ocean Sci. Discuss., doi:10.5194/os-2016-97, 2017 Manuscript under review for journal Ocean Sci. Published: 18 January 2017 c © Author(s) 2017. CC-BY 3.0 License.


Introduction
Internal waves are present throughout earth's oceans wherever there is stratification, from the shallowest near-shore waters to the deepest seas. Internal waves are important to physical oceanographers because they transport momentum and energy, horizontally and vertically, through the ocean, e.g. Munk (1981), Gill (1982). They provide shear to turbulence which results in energy dissipation and vertical 5 mixing, e.g. Holloway (1984), Sandstrom & Elliott (1984). Biological oceanographers are interested because the internal waves carry nutrients onto the continental shelf and into the euphotic zone, e.g. Shea & Broenkow (1988), Sandstrom & Elliott (1984), and Holloway et al. (1985). They are of interest to geological oceanographers because the waves produce sediment transport on the shelf, e.g. Cacchione & Drake (1986). Civil, hydraulic and ocean engineers are interested because the internal waves generate 10 local tidal and residual currents, e.g. Willmott & Edwards (1987), which can cause scour on nearshore as well as offshore structures, e.g. Osborne et al. (1978). Large nonlinear IWs are also of interest to the navy because they cause large vertical displacements and large vertical velocities that may affect underwater operations.
This study is focused on the internal tide and subsequent evolution of nonlinear waves. Internal As the internal tide shoals, the nonlinear terms in the Navier-Stokes equations become important.
These tidal waves of finite amplitude may evolve into packets of high frequency nonlinear waves. The equations describing these waves are much more complex than the linear equations and few mathematical solutions have been found.
We are interested in nonlinear internal waves because they are a very energetic part of the signal 5 in time series that we have observed on continental shelves and in the shallow ocean. We are guided by numerical solutions of Korteweg-de Vries (KdV) type equations that incorporate both weak nonlinear and weak dispersive effects.
The state of the art on the evolution of internal solitary waves across the continental shelf is reviewed in Grimshaw et al. (2010). Grimshaw et al. (2004) simulated the transformation of internal 10 solitary waves across the North West shelf of Australia, the Malin shelf edge, the Arctic shelf; Holloway (1987) discussed the evolution of the internal tide in a two-layer ocean on the Australian North West Shelf. Our model simulations of the evolution of the internal tide across realistic in the Middle Atlantic Bight topography cases are unique since these waves have never been modelled across such topography and stratifications, but the model results are compared with observations made at moorings off the site of the Coastal Mixing and Optics experiment (CMO) in section 4. A summary and conclusions are presented in section 5.

Theoretical Background
The Korteweg de Vries (KdV) equation is well known to be a suitable physical model for 5 describing weakly nonlinear advective effects and linear dispersion in internal waves. It was originally developed by Benney (1966) and extended to second order by Lee & Beardsley (1974). The KdV equation is derived from classical nonlinear long wave theory using a two-parameter perturbation expansion in  and  which scale the nonlinear and dispersive effects, respectively.
The KdV equation, derived following the procedure of Lee & Beardsley (1974) and the discussion 10 by Lamb & Yan (1996), but without mean current, is given by where  is the vertical displacement amplitude of the wave mode, c is the linear long wave phase speed for the mode whose amplitude is ,  and  are coefficients of the non-linear and dispersive terms, while subscripts represent derivatives in time, t, and space, x, respectively. 15 Progressing to 2 nd order in  and  (nonlinear and dispersive effects) yields four additional terms to Eq.(1) -a cubic nonlinear term, as well as higher-order linear and nonlinear dispersive terms -and is known as the fully extended KdV equation (feKdV). Often only the second-order nonlinear term ('cubic nonlinearity') is added resulting in the extended KdV (eKdV) equation

 
where 1 is the coefficient of the cubic nonlinear term. In a two-layer model, for example, when the layers are of similar depth, or when the quadratic nonlinear term is small, the higher-order linear and nonlinear dispersive terms can be omitted, see discussion in Grimshaw et al. (2002). Continuous stratification can support an infinite number of modes. For simplicity we consider wave propagation in a two-layer stratification which supports one mode only. The justification for making this approximation is that most 5 of the energy in the ocean appears to be contained in the first mode anyway, e.g. Alford & Zhao (2007) and discussion therein, while the shelf often has the appearance of a two-layer stratification: an upper mixed layer separated from a weakly stratified bottom layer by a thin pycnocline. This approximation greatly simplifies the problem; the numerical scheme is much less complex for the two-layer case than the continuously stratified case, and the results are easier to interpret. The coefficients of the KdV and 10 eKdV equations are greatly simplified for a two-layer fluid and are written (e.g. Ostrovsky & Stepanyants, 1989) The transformed eKdV is then   0 1 compared to s, terms such as are neglected relative to . The transformed KdV equation is the same as the transformed eKdV equation with 1 = 0. Important solutions of the KdV and eKdV equations are waves of permanent form. One family of these waves are the solitary waves. There is a strong tendency for a long but otherwise arbitrary initial condition to evolve into a train of solitary waves (e.g. Lee & Beardsley, 1974;Drazin & Johnson, 1989).
We are interested in modeling the evolution of the internal tide as it propagates shoreward from the shelf break. Since the greatest oceanic signal is the first internal mode, the stratification of the continental shelf/slope region is modeled as a two-layer fluid. This approximation greatly simplifies the problem; the numerical scheme is much less complex for the two-layer case than the continuously stratified case, and the results are easier to interpret. Using the two-layer model, we study the 15 propagation of the internal tide over various types of topography, including the simplest case of flat bottom with level interface and progressing to realistic topography with sloping interface. All cases have been run within the quadratic nonlinear framework of the KdV equation, and the results are compared with the eKdV model.
For the KdV and eKdV Eqs. (1,2) to be valid, the leading two terms must constitute the primary balance. The nonlinear and dispersive terms can become important, but the assumptions leading to the KdV and eKdV equations are violated if either of the nonlinearity or dispersion terms approach the magnitude of the leading terms. Nonlinear transformation of the internal tide leads to the generation of nonlinear waves which tend to become solitary-like in form as the dispersive term becomes important. 5 We begin by discussing the coefficients of the KdV and eKdV equations for a two-layer fluid, where the density difference between the layers is chosen to be a constant: g/ = .014 m/s 2 , a representative value for the Coastal Mixing and Optics (CMO) experiment (Levine & Boyd, 1999), for example at a mooring in the Middle Atlantic Bight located at 40.5 o N, 70.5 o W, and also in agreement with the stratification near the mooring location displayed in Barth et al. (1998). The linear phase speed, c, is is contained within the phase speed (Apel, 1987) and lines of constant total water depth are perpendicular to the line h1=h2. For a given total water depth, the speed is greatest when h1=h2 and decreases as 15 difference in layer thickness increases. Starting at a point on the line h1=h2 and keeping the thickness of one of the layers constant, the speed of the wave decreases as the thickness of the other layer decreases.
The coefficient of the non-linearity term, , is also a function of h1 and h2 only, Eq. 3b and layer decreases. The absolute value of /c changes least rapidly when h1 h2. When the thicker layer is larger than the thinner layer by at least a factor of 2-3, then /c is relatively insensitive to the thickness of the thick layer, that is when h2>>h1, then |/c|  3/2h1 and is not a function of h2. /c is also important since when multiplied by the amplitude, , it represents the ratio of the nonlinear to the linear terms in the KdV Eq. (1). 5 The coefficient of the dispersive term, , divided by c is also a function of h1 and h2 only, Eq.
3c and Fig. 2c, whose values are symmetric about the line h1=h2. The value of /c for any given water depth is a maximum when h1=h2, values decrease as either of the layers becomes small. The interpretation of Fig.2c as a ratio of terms is complicated since, unlike Fig.2b, the derivatives do not cancel and the ratio cannot be simplified. 10 The coefficient of the cubic nonlinear term, 1, when divided by c is also a function of h1 and h2 only, Eq. (3d) and Fig. 2d. 1 is always negative and is symmetric around the line h1=h2, while for a given water depth the magnitude of 1 is least when h1=h2. The value of 1 as either one of h1 or h20.
It is also useful to calculate the ratio /1, see O'Driscoll (1999). The relative importance of the quadratic to cubic nonlinearity is given by /1. For a given water depth cubic nonlinearity is most important 15 when h1  h2, i.e. when the magnitude of  is small. The magnitude of the quadratic nonlinear term is much greater than that of the cubic nonlinear term when the water depth of one layer is much greater than the other and in this case the eKdV model is very similar to the KdV model.

The Korteweg de Vries (KdV) Model solutions
Using the KdV equation, we first investigate 4 cases with level bottom for different combinations of h1 and h2. We then progress to constant sloping bottom, with both horizontal and sloping interface.
Finally, we make model runs with realistic topography at the sites of the CMO.

Level Bottom
We begin by studying the evolution of the internal tide over a level bottom, with level interface (h1 and h2 constant). This simple fluid arrangement is instructive when developing an intuitive feel for the generation and propagation of internal wave packets. A level bottom is also a good approximation for the continental shelf where the total water depth changes slowly in the horizontal. Four cases (Cases 1-4) 10 using different layer thickness were selected to look at the effects of different relative magnitudes of  and  (Table 1) 15 The leading waves at the left have the shape expected for a sech 2 solitary wave (Fig. 4a). The trailing waves to the right appear more sinusoidal in shape, and are relatively more dispersive than a sech 2 wave.
Upwards of twenty waves have formed when the internal tide has traveled 160 km. The leading six to seven waves travel with speed greater than c and have a nearly sech 2 form. The trailing waves travel slower than c as expected for waves that are more dispersive. For Case 2 we choose h1 and h2 such that /c = .02 as in Case 1, but the value of /c is less than half that of Case 1. Since the ratio of the dispersive coefficient to the nonlinear coefficient has been reduced by more than half, we expect Case 2 to be more nonlinear, the internal tide to steepen sooner, and nonlinear internal waves to form at smaller l. Fig. 5 shows that the internal tide evolves similarly to the remaining waves have speed less than c, and disperse from the leading waves as l increases. 10 For Case 3 we choose /c = 1250, as in Case 1, but with /c = .0021, a factor of ten less than the value used in Case 1 and 2. As a result, we expect the internal tide to be much less nonlinear. Indeed, the internal tide steepens slowly and even by l = 200 km solitary-type waves have not been generated (see O'Driscoll 1999).
For Case 4 the nonlinearity parameter is half that used in Case 1, /c = .01 and /c is the same 15 value. So, we expect the resultant internal tide to be more nonlinear than Case 3 but less so than either of

Constant Bottom Slope
The propagation of the internal tide along constant sloping topography was studied for cases of constant upper layer thickness (Case A) and sloping interface ( Fig.8b where the leading waves are compared with sech 2 solitary form. Beyond 100km the waves 10 rapidly approach solitary waves of elevation since  becomes large quickly. As the internal tide propagates into shallow water the leading face of the wave steepens but, unlike cases 1-4, the decreasing magnitude of  causes this steepening to slow down and there is virtually no change in wave slope steepness between 70 and 90 km. The rate of change of the slope of the leading face changes sign when  becomes positive and the slope steepens rapidly, while the back face of the 15 internal tide slackens. The steepening of the leading wave will lead to the formation of a second shocklike front (or a "reverse hydraulic jump" as has been described by Holloway et al., 1997). The leading waves are slightly more nonlinear than dispersive when l 70 km but become less so as l approaches 100 km. When =0 (l=100km) the value of the nonlinear term is zero and the waves look like a dispersive packet. Since  >0 for l >100 km, the nonlinear term is again a factor and the waves  5 We now proceed to the transformation of the internal tide for the case of realistic topography for the CMO site.

Realistic topography and stratification
The CMO site was located in the Middle Atlantic Bight. CTD profiles were made across the continental shelf from shallow water to beyond the continental slope. Boyd et al. (1997) have concluded that the internal tide at the site is primarily a first mode internal wave, further justifying our choice of a two-layer slightly greater than c, as in Case A. The waves slow down to travel at speed c where l  90km and  is very small. The speed of the waves then becomes slightly slower than c but faster and more complicated 15 than Case A, due to the undulating topography.
The difference in magnitudes of the nonlinear and dispersive terms, , is plotted in Fig. 11. The leading 2-3 three waves are initially more nonlinear than dispersive but the diminishing magnitude of  leads to the waves becoming more dispersive-like and the waves begin to slow down. The negligible value of  between l=100-115km results in the waves behaving very much like a dispersive packet and 20 they travel with wave speed slightly less than c. The increasing value of  after it passes through zero, leads to the nonlinear term becoming almost the same order of magnitude as the dispersive term before the model becomes numerically unstable shortly beyond l=130km.

The extended Korteweg -de Vries (eKdV) model
All of the model runs discussed in section 3.1 were also made using the extended Korteweg-de The KdV and eKdV models are so different at the CMO site when compared to Case A because the magnitude of 1 is greater at the CMO site. Though the magnitude of  is less in Case A, the fact that the magnitude of 1 is so small when compared to  means the addition of the cubic nonlinear term does 5 little to change the KdV results. This is not true at the CMO site where the greater magnitude of 1 is the reason for the difference between the KdV and eKdV frameworks, particularly as the internal tide propagates into shallower water and the magnitude of the ratio /1 is much greater for Case A.
Comparing the leading waves from the eKdV and KdV solutions reveals a fundamental difference in wave form; the KdV waves are taller and thinner (Fig. 13c). Solitary type solutions to the KdV (sech 2 ) 10 and to the eKdV (tanh) are fitted to the leading waves ( Fig. 13d-e). The leading wave in the KdV model is very well approximated by a sech 2 wave. The lead wave in the eKdV model is neither well approximated by sech 2 or tanh, but appears to be a hybrid between the two. Fits of sech 2 and tanh waves were made by subjectively choosing values of 0 and , respectively, while using the value of KdV and eKdV parameters for 69m water depth. Note that the amplitude of the tanh wave is limited to /1. 15 Increasing  only serves to make the waves wider once the value of  is close to one (Fig. 1c). The amplitude and width of the leading waves of the packet are also compared in Fig. 14a. The width is defined as the time it takes the wave to pass a fixed point, as measured at 42% of the amplitude. Results To learn more about the evolution of a sine wave to waves with sech 2 and tanh form, we ran the model with constant parameters (flat bottom) using values at the mooring site. The runs were made with 10 initial tidal amplitudes of 1, 2 and 4 m in both KdV and eKdV frameworks and the width vs. amplitude for the first and second wave in each packet is plotted at various increments of l (Fig. 14b). The KdV waves grow in amplitude with approximately constant width before turning to hug the theoretical KdV line. They then decrease in amplitude while increasing slightly in thickness. Though the KdV model waves continue to evolve, most of them can be well approximated as being 'sech 2 ' waves after ~100km 15 (as was previously shown for Case 1 and Case 4). For the eKdV case, the waves are initially close to the theoretical sech 2 KdV curve. The waves move slowly towards the theoretical eKdV tanh curve, ultimately decreasing in amplitude and increasing in thickness. The last points have been plotted after the internal tide has propagated ~240km. It appears that these waves are evolving toward tanh form, but mature over a long distance. Also, the amplitudes of the waves are greater than the theoretical eKdV maximum but 20 their magnitudes decrease as the tide evolves. Another investigation to explore the evolution in the eKdV model (constant parameters) was made using an initial condition of a sech 2 wave, the solitary wave solution to the KdV equation. Sech 2 amplitudes of 4m, 7m, 9m, and 13m (Fig. 14c) were chosen. The sech 2 waves are rapidly transformed to tanh waves, e.g. the 4 examples plotted reach the theoretical eKdV curve after the wave has propagated about 10 km. A solitary sech 2 wave evolves much more rapidly to the tanh form (Fig. 14c), as opposed to 5 when it is part of a packet of waves (Fig. 14b). The reason for this has not been thoroughly investigated, but provides caution for treating a packet as a group of non-interacting waves.

Observations of Nonlinear Internal Waves
The data to be presented and discussed was collected during the CMO, for location see Fig. 15. The CMO 10 experimental field program was conducted to increase our understanding of the role of vertical mixing processes in determining the mid-shelf vertical structure of hydrographic and optical properties. The field program was conducted on a wide shelf so as to reduce the influences of shelf break and nearshore processes. The data we discuss was collected from the CMO Central Mooring in July and August 1996, a time when a strong thermocline is present as a result of large-scale surface heating, Boyd et al. (1997).

Observations during the Coastal Mixing and Optics Experiment
The series for the period 29 July to the 31 August 1996 (year day 210 -245, Fig. 16a). The dominant barotropic tidal signal in the Middle Atlantic Bight is semi-diurnal, and is strongest over the period day 241-245 during spring tide (Fig. 17). A semi-diurnal signal is apparent in the first mode record, particularly during the spring tide period. A spectrum of the first mode amplitude (Fig. 18) shows energy peak at both low and high tidal frequencies. Much of the high frequency energy is due to bursts or pulses of high frequency 5 nonlinear internal waves that occur for a short period during the semi-diurnal tidal cycle. These nonlinear internal waves propagate shoreward across the continental shelf to the south of Martha's Vineyard. The energy at high frequency is greater over the period day 241-245 during spring tides (Fig. 18). There is a clear maximum in energy at 2 cpd over this period, and a significant amount of energy is also contained at 4 cpd. The energy rapidly drops for frequencies greater than 4 cpd but there is a significant increase in 10 energy at ~50 cpd and at ~90 cpd. To help interpret these observations, we compare them with the twolayer eKdV model using the CMO parameters. Since we do not know where the internal tide is generated or its amplitude, the model was run assuming a sinusoidal internal tide at distances of 24 km, 48 km and 60 km seaward of the mooring site. Three initial amplitudes of 2 m, 4 m and 6 m were used at each distance. Fig. 14c shows the internal tide as it appears at the CMO mooring site for these nine cases. In 15 all cases, the leading face of the periodic sinusoidal wave slackens (or flattens) as the internal tide propagates shoreward. This is followed by a steepening of the back face which develops into a shocklike front. The shock-like front is followed by oscillations which subsequently evolve into a packet of solitary-like waves.
This same pattern can often be seen in the observed time series of the first internal mode. Fig. 19 20 shows several individual jumps at the CMO mooring. Fig. 19a  match the model results of Fig.16c. Some features of the observations compare well with the model. The slackened leading face of the tide is always followed by a steep -almost shock like -front followed by several highly nonlinear short period waves. Although not rank ordered, the largest amplitude wave in the observed packet is always at or near the jump. The model results show that the amplitude of the jump is greater for larger initial condition, and decreases with distance from the point of generation. Although 5 nonlinear waves continue to evolve, their amplitudes decrease as they propagate shoreward from their generation point, and they become 'thicker', i.e. they become more tanh like. Though the modelled waves have amplitudes less than the theoretical tanh limit for local eKdV parameters, they nonetheless fit the shape of several observed waves at the CMO site.
There are also features of the observations that are not found in the model. Fig. 19 (left panel f 10 and g) differ in that the packet that follows the shock-like front, persists until the end of the tidal period, and the waves are spread apart from each other. Fig. 19 (left panel c) shows two packets of solitary-like waves propagating past the mooring site over a tidal period. The leading slackened face is followed by a shock-like front and a packet of solitary waves. The trailing face then slackens to assume a slope similar to the leading face but a second shock-like front, followed by a packet of solitary waves, passes before 15 the end of the tidal period. This could be from a second internal tide front coming from another generation site, there can be overlapping semi-circles of internal wave fronts from multiple generation sites, see for example discussion in Apel et al. (1988).
Another common observation that is not found in the model results is a 'drop' in amplitude before the jump that occurs at the beginning of the wave packet. Fig. 19 (left panel h) shows that the 20 first internal mode drops between day 243.5 and 243.6 but the slackening slope is restored before the 25 Ocean Sci. Discuss., doi:10.5194/os-2016-97, 2017 Manuscript under review for journal Ocean Sci. 'thinner' than model waves with similar amplitudes. However, it seems reasonable to say that the observed waves are a good fit to the model waves.
While some features of the observations are reproduced in the model, there are many differences. The eKdV model used here is highly idealized. There are many effects that have not been included, including bottom and internal friction, earth's rotation and mean shear. Given these 5 limitations, we conclude that the observations are reasonably well matched by our model.

Summary and conclusions
Observations faster than the linear wave speed, c, and nearly fit solitary wave form for local KdV parameters ("sech 2 ").
The trailing waves usually travel slower than c, tend to be thinner than the local sech 2 waves and are relatively more dispersive than the leading waves.
The transformation of the internal tide is dependent upon the ratio of the nonlinear to linear terms, All of the model runs made within the KdV framework were also made within the eKdV framework which includes a cubic nonlinearity term scaled by 1. The results may or may not be similar, depending 15 upon the ratio of the two nonlinear terms, To better understand the evolution of waves toward tanh form in an eKdV framework, without the complications of varying parameters, model runs were made using constant eKdV parameters representative of the CMO site. Upon formation, the leading waves of the packet are similar to sech 2 waves. 5 The waves become 'thicker' and tend toward the tanh form upon further propagation, but never reach the theoretical tanh curve in our limited domain. To help understand why the evolution of waves from being close to sech 2 waves to being close to tanh waves was so slow, the internal tide was forced with a sech 2 wave. The evolving sech 2 rapidly moves to the theoretical tanh curve for all amplitudes. We conclude that the interaction between the solitary like-waves in a packet slows them from evolving into exact solitary 10 'sech 2 ' or 'tanh' waves.
Model runs with varying initial amplitudes and generation regions were made to help interpret the observations made at the CMO site. Some features of the observations compare well with the model. The leading face of the internal tide steepens to form a shock like front. Nonlinear high frequency waves evolve shortly after the appearance of the jump. Although not rank ordered, the wave of maximum amplitude is 15 always close to the jump. Some features of the observations are not found in the model. Nonlinear waves can be very widely spaced and persist over a tidal period. The amplitude of the observed waves often decreases before the arrival of the jump, while the leading face may change slope before the jump arrives. waves evolving from large amplitude model waves. A large fraction of smaller amplitude observed waves, particularly less than 10 m, are thinner than model waves of similar amplitude. We conclude that the observed waves are a good match to modeled waves given the highly idealized eKdV model used, and the fact that we have neglected friction, rotation and mean shear.

Author Contribution
Kieran O'Driscoll conducted this work out while a graduate student at the College of Oceanic & Atmospheric Sciences, Oregon State University, in partial fulfillment of the degree of Master of Science.
Murray Levine was the student's advisor. and 13 m. A value is plotted every 1 km up to a maximum distance of 15 km. The theoretical width vs. 15 amplitude for sech 2 and tanh waves is also shown (dotted lines). The width is calculated at 42% of the total amplitude.